Atlas › Modular forms › Weight 2 › Level 931 › 931.2.a

Newspace 931.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(931))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
931.2.a.a	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} - 3q^{5} + O(q^{8})$	+	931b1 rank 0
931.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 4q^{6} + O(q^{8})$	+	931c1 rank 0
931.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 4q^{6} + O(q^{8})$	−	931a1 rank 0
931.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{3} + 3q^{4} - 7q^{6} + O(q^{8})$	−
931.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{6} + O(q^{8})$	−
931.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} - 4q^{6} + O(q^{8})$	+
931.2.a.g	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{3} + 3q^{4} + 6q^{5} + 5q^{6} + O(q^{8})$	+
931.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{6} + O(q^{8})$	−
931.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{6} + O(q^{8})$	+
931.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 3q^{3} - q^{4} - 2q^{5} + q^{6} + O(q^{8})$	−
931.2.a.k	3	degree 3	$3q + 2q^{2} - 3q^{3} + 6q^{4} + 2q^{5} + q^{6} + O(q^{8})$	+
931.2.a.l	4	degree 4	$4q - 2q^{3} + 2q^{4} - 8q^{5} + 4q^{6} + O(q^{8})$	−
931.2.a.m	4	degree 4	$4q + 2q^{3} + 2q^{4} + 8q^{5} - 4q^{6} + O(q^{8})$	+
931.2.a.n	7	degree 7	$7q + 2q^{2} - 2q^{3} + 10q^{4} - 2q^{5} - 4q^{6} + O(q^{8})$	−
931.2.a.o	7	degree 7	$7q + 2q^{2} + 2q^{3} + 10q^{4} + 2q^{5} + 4q^{6} + O(q^{8})$	+
931.2.a.p	10	degree 10	$10q - 2q^{2} - 4q^{3} + 10q^{4} - 16q^{5} - 8q^{6} + O(q^{8})$	+
931.2.a.q	10	degree 10	$10q - 2q^{2} + 4q^{3} + 10q^{4} + 16q^{5} + 8q^{6} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
931.2.a.a	$x - 6$
931.2.a.b	$x + 4$
931.2.a.c	$x - 8$
931.2.a.d	$x^{2} - 6x + 4$
931.2.a.e	$x^{2} + 2x - 31$
931.2.a.f	$x^{2} + 2x - 7$
931.2.a.g	$x^{2} - 8x - 36$
931.2.a.h	$x^{2} - 8$
931.2.a.i	$x^{2} - 32$
931.2.a.j	$x^{2} + 8x - 4$
931.2.a.k	$x^{3} + 7x^{2} - 34x - 236$
931.2.a.l	$x^{4} + 6x^{3} - 62x^{2} - 344x + 244$
931.2.a.m	$x^{4} - 6x^{3} - 38x^{2} + 304x - 476$
931.2.a.n	$x^{7} + 4x^{6} - 154x^{5} - 278x^{4} + 6693x^{3} + 3774x^{2} - 87588x + 36744$
931.2.a.o	$x^{7} - 8x^{6} - 154x^{5} + 1138x^{4} + 4485x^{3} - 21318x^{2} - 67212x - 34248$
931.2.a.p	$x^{10} + 14x^{9} - 73x^{8} - 1476x^{7} + 306x^{6} + 46304x^{5} + 41922x^{4} - 525416x^{3} - 716952x^{2} + 1765840x + 2795708$
931.2.a.q	$x^{10} - 10x^{9} - 169x^{8} + 1644x^{7} + 8466x^{6} - 68680x^{5} - 225150x^{4} + 784600x^{3} + 3006216x^{2} + 2018704x - 104164$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(931))$	62
Old forms	23
Cusp forms $S_{2}(\Gamma_0(931))$	85
Eisenstein series $E_{2}(\Gamma_0(931))$	15
Modular forms $M_{2}(\Gamma_0(931))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(931)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(133))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.312 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 7, 19, 49, 133, 931]: [0, 1, 2, 5, 14, 94] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(931, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 931:2 1000

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